Equation (REF ) has been studied by various researchers, including [1]}, [2]}, [3]}. The term in brackets on the RHS of (REF ) that is quadratic in \(U^\alpha \)  arises from the transformation of derivatives with respect to \(s\) , to derivatives with respect to \(\tau \) , and using (REF ). This is indeed the form expected for a scalar field force.[4]}
